Global wellposed problem for the 3-D incompressible anisotropic Navier-Stokes equations

In this paper, we consider a global wellposed problem for the 3-D incompressible anisotropic Navier-Stokes equations ($\mathit{\text{ANS}}$). In order to do so, we first introduce the scaling invariant Besov-Sobolev type spaces, $B_p^{-1+\frac{2}{p},1\mathrm{/}2}$ and $B_p^{-1+\frac{2}{p},1\mathrm{/}2}(T)$, $p\ge 2$. Then, we prove the global wellposedness for ($\mathit{\text{ANS}}$) provided the initial data are sufficient small compared to the horizontal viscosity in some suitable sense, which is stronger than $B_p^{-1+\frac{2}{p},1\mathrm{/}2}$ norm. In particular, our results imply the global wellposedness of ($\mathit{\text{ANS}}$) with high oscillatory initial data.